That's a surprisingly difficult question to answer, and the answer varies from year to year. This year, the key differences are: the Classic has fancier electronics, with 4 push/pull pots for pickup configuration and the Standard bursts have flame tops. Necks, pickups, tuners are all the same. Although there's a '50's version of the Standard that has a fatter neck. Also, the classic has weight relief and the standard does not, so the standards weigh more. They both have a mahogany body with binding and a maple cap and a rosewood fretboard.
Generally speaking, the Standard is intended to be more of a classic model, while the Classic is intended to be more modern, but don't confuse that with the Les Paul Modern model. It's confusing
. But I will say the new guy who took over last year is making great strides in putting the company on the right track.